Both equal temperament and just intonation have heretofore been held to be ideal scales that could not be precisely attained in practice. Although the principle of equal temperament has become firmly established in the last 300 years as the standard for musical intonation, few, if any, skilled musicians abide by it, and mechanical instruments, due to inherent physical limitations, cannot maintain it. Hence the assignment of equal intervals, its historical argument for providing freedom of modulation to other tonalities, or signature keys, has no basis in reality.
In the study as well as in the performance of music no subject matter has been more controversial than the centuries-old dispute regarding tempered scales and just intonation. A current example of their tonal differences may be observed in viewing a piano or organ keyboard, where, for instance, C.music-sharp. and D.music-flat. are represented by the same black digital, or manual key, although these notes are written in musical notation as two separate and distinct notes. Moreover, depending upon the tonality, or signature key, in which a musical composition is written, certain white digitals are on occasion assumed to be sharps or flats.
Similar discrepancies of so-called "enharmonic notes," which harmony textbooks describe as being "practically" the same note, exist in virtually all other fixed-frequency instruments, whether plucked or blown, although skilled performers manage to modify the frequency, or "pitch," slightly to bring a note "into tune," generally to conform to a predominating tone, or in consonant relation thereto, when performing with other instruments. Thus a good deal of a performer's artistry depends upon an ability to make perceptible corrections in frequency, where possible. This is part of a learning process. (See, for example, "Intervals, Scales & Temperaments," by Llewellyn S. Lloyd and Hugh Boyle, St. Martin's Press, New York, 1979, pp. 280-286.)
No less significant, but generally overlooked, is the fact that when modulation, or transition to other tonalities, is called for in written music, several of the notes of one tonality require small but distinct changes in frequency for the same nominal, or written, notes of the previous tonality. These changes are known as "syntonic commas," or ratios of 81/80 for each increasing number of sharps and 80/81 for every added flat in the key signatures. Whereas equal temperament is based on two intervals, a semitone and a whole tone equal to two semitones, just intonation has three basic diatonic intervals: a semitone (16/15), a minor tone (10/9) and a major tone (9/8). Moreover, C.music-sharp. has a ratio of 25/24, but D.music-flat. is 16/15.
For centuries numerous scholars and critical listeners have argued that the influence of fixed-pitch instruments have contributed to a loss of correct pitch and has caused vocalists and instrumentalists not constrained by fixed pitch to sing and play "out of tune" either for equally tempered or "just" performance. Basic to this problem has been the lack of technological development in instruments for either tempered tuning or just intonation. An examination of the abundant literature on the subject discloses that no fixed-pitch or keyboard instruments have previously been proposed or built capable of approaching precisely equal tempered intervals, nor any that could accurately produce just intonation and all of its enharmonic notes or modulational pitch changes for either instructional or performance use.
The degree of perfection implied by the term "equal temperament" was first shown in 1595 by the Chinese prince Chu Tsai-yu, whose recorded monochord string lengths show no greater semitone error than two-millionths cent when converted into present-day logarithmic calculations of 12 equal semitones totaling 1200 cents, or 100 cents for each semitone. Yet the tuning of pianos and organs has seldom yielded accuracies of better than two cents.
Just intonation, as applied to the development of harmonic needs, has not been so well defined, although the precise diatonic interval relationships were clearly given by Claudius Ptolemy in about 200 A.D. More than a millennium passed before flats were firmly established in music, and sharps appeared in the 17th-Century emergence of opera as an artistic form. These so-called "accidentals" in the historical progression from strictly melodic to freely harmonic music led to many compromise scale systems, but these invariably fell short of meeting either tonal needs or playable keyboards.
Helmholtz, in his monumental treatise, "On the Sensations Of Tone As A Physiological Basis For The Theory Of Music," translated by A. J. Ellis and published in 1875 by Longmans, Green & Co, London, gave his appraisal of the practicability of just intonation in a keyboard instrument, when he said (p.491): " . . . In order to obtain perfect intervals in all keys, it would certainly be scarcely possible to overcome the difficulties of the problem." Regarding specific keyboards, Helmholtz reported (p. 499) that Praetorius, a 17th-Century annotator, had seen in Prague an instrument having 19 digitals to the octave, "the black digitals being doubled, and others inserted between those for e and f, and between those for b and c," but the tuning was applied to an improved temperament, not to a just-intonation scale.
All of the known proposals for solving the tonal or keyboard problems were merely reflections of what one or another inventor thought to be adequate for representing written music correctly. (See, for example, "Enharmonic Key-Board for Organs & c" of H. W. Poole, U.S. Pat. No. 73,753, for scales "in every key or signature.") The efforts of numerous innovators gave rise to other intervals anomalous to the Ptolemaic relationships (i.e., those variously referred to as natural, exact, right, pure, true, correct, or just), some of them based on extensions of the harmonic series of a single tone, others upon a variety of "cyclic" approximations and still others on Pythagorean (3/2-related) ratios, which last properly apply to modulation but not to just-intonation intervals within a single tonality as perceived by the ear in harmony or in chordal relationships.
In more recent years just intonation has become largely a curiosity as well as a perplexity to music students who were taught the differences of enharmonic notes, were asked to disregard them for purposes of studying harmony, and yet were expected to produce the correct intonation on instruments not constrained by fixed pitch, or in singing. Such instrumentalists and vocalists have been accused of false intonation when their pitch did not exactly correspond to a fixed-pitch instrument, however incorrectly tuned. Moreover, modulaton, as a transition to a related tonal center, has been reduced to a form of fixed, keyboard-note transposition as dictated by the tuning of an equally tempered scale of questionable accuracy.
Significantly, just intonation was taught as a system in sight-reading of mucic to English schoolchildren by means of a movable-Do method of modulation known as Tonic Sol-fa, introduced by Sarah Ann Glover in the early 19th Century and developed by John Curwen into a national movement. (See "Tonic Sol-fa," Encyclopedia Britannica, Vol. 22, pp. 283-284, 1958.) By contrast, the Solfeggio system, taught in music schools and conservatories, used a fixed-Do method, an outgrowth, particularly in Italy and France, of the influence of fixed-pitch instruments, dating from the 18th Century. Its stated purpose was to encourage the development of pitch memory in the teaching of sight-singing to conform with tempered tuning.